Problem 55 - Algebra and Trigonometry

Question

Find each sum. The sum of the first 120 terms of the sequence $$ 14,16,18,20, \ldots $$

Step-by-Step Solution

Answer

15960

Step 1 - Identify the sequence type

The given sequence is an arithmetic sequence because each term increases by a constant difference. The first term is 14 and the common difference is 2.

Step 2 - Write the formula for the sum of an arithmetic sequence

The formula for the sum of the first n terms of an arithmetic sequence is: \[ S_n = \frac{n}{2} (a_1 + a_n) \] where \( a_1 \) is the first term, \( a_n \) is the nth term, and \( n \) is the number of terms.

Step 3 - Find the 120th term

To find the 120th term (\( a_{120} \)), use the formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n-1)d \] Substitute \( n = 120 \), \( a_1 = 14 \), and \( d = 2 \): \[ a_{120} = 14 + (120-1) \times 2 = 14 + 238 = 252 \]

Step 4 - Calculate the sum of the first 120 terms

Substitute \( n = 120 \), \( a_1 = 14 \), and \( a_{120} = 252 \) into the sum formula: \[ S_{120} = \frac{120}{2} (14 + 252) = 60 \times 266 = 15960 \]

Key Concepts

arithmetic sequence

An arithmetic sequence is a simple pattern of numbers where each term increases by a fixed amount called the common difference. Think of it as a line of dominos where each domino stands a set distance from the next. For example, in the sequence 14, 16, 18, 20, each number is 2 units more than the previous one, making 2 the common difference. The starting number (14 in this example) is known as the first term.

sum formula

To find the sum of terms in an arithmetic sequence, we use a specific sum formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \] This formula might look complicated at first, but it's pretty straightforward. Let's break it down:

• \( S_n \) is the sum of the first \( n \) terms.
• \( n \) is the number of terms you want to add up.
• \( a_1 \) is the first term in the sequence.
• \( a_n \) is the nth term.

By plugging in these values, the formula helps you quickly calculate the sum of the sequence without adding each term individually.

sequence terms

In an arithmetic sequence, each term is generated by adding the common difference to the previous term. To find any term at position \( n \) (known as the nth term), we use this formula: \[ a_n = a_1 + (n-1)d \] Where:

• \( a_n \) is the nth term you're trying to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term's position in the sequence.
• \( d \) is the common difference between consecutive terms.

For example, to find the 120th term of 14, 16, 18, 20,..., you would set \( a_1 = 14 \), \( n = 120 \), and \( d = 2 \). The calculation would be: \( a_{120} = 14 + (120-1) \times 2 = 252 \) Once you know the nth term and other values, you can easily use the sum formula to find the total of all terms up to \( n \). Happy calculating!